On elec­tion night, un­of­fi­cial re­sults showed Mr. Scott had beaten Mr. Nel­son by a bit less than 60,000 votes — the same ra­zor thin mar­gin by which Mr. Scott had been re-elected gov­er­nor in 2014.

But as of­fi­cials in the deeply lib­eral Broward and Palm Beach coun­ties an­nounced more and more votes Wed­nes­day and Thurs­day — which whit­tled away at Mr. Scott’s lead — and squads of Demo­cratic lawyers de­scended on Flor­ida promis­ing re­counts and law­suits, the Scott cam­paign and some Repub­li­can of­fi­cials cried foul.

The vote to­tal in Broward County, which is no stranger to du­bi­ous vot­ing pro­ce­dures, stood at 634,000 votes on elec­tion night, ac­cord­ing to the Scott cam­paign. That fig­ure swelled to 695,700 votes by 1 a.m. Thurs­day and then to 707,223 votes by 2:30 p.m.

“And we just learned, that the num­ber has in­creased to 712,840 bal­lots cast on elec­tion day,” the Scott cam­paign said at 8 p.m. EST Thurs­day.

And all of this has un­folded against a back­drop of si­lence from county of­fi­cials about where the thou­sands of votes had been or how many more they may claim to need count­ing.

“Demo­crat lawyers are de­scend­ing on Flor­ida,” Mr. Ru­bio tweeted Thurs­day. “They aren’t here to make sure ev­ery vote is counted. They are here to change the re­sults of the elec­tion; & #Broward is where they plan to do it.”

Mr. Scott also did not mince words Thurs­day evening.

“We’ve all seen the in­com­pe­tence and ir­reg­u­lar­i­ties in vote tab­u­la­tions in Broward and Palm Beach for years. Well, here we are again,” he said in a state­ment re­leased by his cam­paign. “I will not sit idly by while un­eth­i­cal lib­er­als try to steal this elec­tion from the peo­ple of Flor­ida.”

Even Pres­i­dent Trump weighed in Thurs­day evening, say­ing on Twit­ter that “Law En­force­ment is look­ing into an­other big cor­rup­tion scan­dal hav­ing to do with Elec­tion Fraud in #Broward and Palm Beach. Flor­ida voted for Rick Scott!”

Marc Elias, a lawyer with the Demo­cratic-con­nected firm Perkins Coie, held a con­fer­ence call with re­porters Thurs­day morn­ing at which he brashly pre­dicted Mr. Nel­son would even­tu­ally be de­clared the vic­tor in the race and boasted of his track record in flip­ping the re­sults of elec­tions that al­lowed Demo­cratic can­di­dates who ini­tially looked like losers to claim vic­tory in the end.

Mr. Scott and his “bil­lions of dol­lars” wouldn’t be able to stop it, Mr. Elias said.

In re­sponse, the Scott cam­paign said Mr. Elias had been dis­patched “to steal the elec­tion,” with the con­nivance of Demo­cratic of­fi­cials and lawyers who Thurs­day ap­peared to dic­tate the puz­zling growth in the num­ber of bal­lots.

“Ev­ery day since the elec­tion, the left-wing ac­tivists in Broward County have been com­ing up with more and more bal­lots out of nowhere,” Mr. Scott’s state­ment read. “We all know what is go­ing on. Ev­ery per­son in Flor­ida knows ex­actly what is hap­pen­ing. Their goal is to keep mys­te­ri­ously find­ing more votes un­til the elec­tion turns out the way they want.”

The Democrats’ vow to pur­sue le­gal ac­tion is the sec­ond stage, ac­cord­ing to Mr. Scott.

“And when that fails, they will file a bunch of law­suits in or­der to try to over­turn the will of the vot­ers,” he said.

Dan McLaugh­lin, a spokesman for Mr. Nel­son, ac­cused Mr. Scott of “po­lit­i­cal mo­ti­va­tion” in his law­suit.

“The goal here is to see that all the votes in Flor­ida are counted and counted ac­cu­rately,” he said in a state­ment. “Rick Scott’s ac­tion ap­pears to be po­lit­i­cally mo­ti­vated and borne out of des­per­a­tion.”

The fren­zied post-elec­tion ac­tiv­ity in Flor­ida on the Se­nate race has also trick­led down to the gu­ber­na­to­rial con­test. There, for­mer GOP Rep. Ron DeSan­tis beat Demo­cratic Tal­la­has­see Mayor An­drew Gil­lum, who con­ceded on Tues­day night.

But as piles of votes were de­clared found and counted for Mr. Nel­son, Mr. DeSan­tis’ lead also di­min­ished and Mr. Gil­lum re­leased a state­ment sug­gest­ing he may re­scind his con­ces­sion.

“On Tues­day night, the Gil­lum for Gov­er­nor cam­paign op­er­ated with the best in­for­ma­tion avail­able about the num­ber of out­stand­ing bal­lots left to count,” the state­ment read. “Since that time, it has be­come clear there are many more un­counted bal­lots than was orig­i­nally re­ported. Our cam­paign, along with our at­tor­ney Barry Richard, is mon­i­tor­ing the sit­u­a­tion closely and is ready for an out­come, in­clud­ing a state­man­dated re­count.”

“As more votes are be­ing counted, Sen. Bill Nel­son’s vote share is in­creas­ing and will con­tinue to in­crease,” one of them read. “We be­lieve Sen. Nel­son will be the win­ner of this elec­tion at the end of this process and we must see it through to the end.”

Flor­ida ma­chine re­counts are trig­gered when the mar­gin be­tween the can­di­dates is be­low 0.5 of 1 per­cent; a hand re­count takes place when that mar­gin di­min­ishes to a quar­ter per­cent.

The Flor­ida sec­re­tary of state has not or­dered a re­count, and his of­fice said that would not hap­pen un­til can­vass­ing boards re­turn their un­of­fi­cial re­turns on Nov. 10.

The ac­tiv­ity is par­tic­u­larly cu­ri­ous given ar­eas that were rav­aged by Cat­e­gory 4 Hur­ri­cane Michael less than a month ago seem to have con­ducted their elec­tions with­out is­sues, whereas deep blue pock­ets in the south­ern part of the state con­tinue to pro­duce votes.

AS­SO­CI­ATED PRESS

Sen. Bill Nel­son, Flor­ida Demo­crat, had al­ready sig­naled his in­ten­tion to fight the elec­tion re­sults of his race against Repub­li­can Gov. Rick Scott in the Flor­ida Se­nate race. The two ran neck and neck in the midterms.